NEAR-SURFACE LOCALIZATION AND SHAPE IDENTIFICATION OF A SCATTERER EMBEDDED IN A HALFPLANE USING SCALAR WAVES

We discuss the inverse problem associated with the identification of the location and shape of a scatterer fully embedded in a homogeneous halfplane, using scant surficial measurements of its response to probing scalar waves. The typical applications arise in soils under shear (SH) waves (antiplane motion), or in acoustic fluids under pressure waves. Accordingly, we use measurements of either the Dirichlet-type (displacements), or of the Neumann-type (fluid velocities), to steer the localization and detection processes, targeting rigid and sound-hard objects, respectively. The computational approach for localizing single targets is based on partial-differential-equation-constrained optimization ideas, extending our recent work from the full-1 to the half-plane case. To improve on the ability of the optimizer to converge to the true shape and location we employ an amplitude-based misfit functional, and embed the inversion process within a frequency- and directionality-continuation scheme, which seem to alleviate solution multiplicity. We use the apparatus of total differentiation to resolve the target's evolving shape during inversion iterations over the shape parameters, à la.2,3 We report numerical results betraying algorithmic robustness for both the SH and acoustic cases, and for a variety of targets, ranging from circular and elliptical, to potato-, and kite-shaped scatterers.

II. On a class of differential equations, including those which occur in dynamical problems

This paper is intended to contain a discussion of some properties of a class of simultaneous differential equations of the first order, including as a particular case the form (which again includes the dynamical equations), x ' i = d Z/ dy i , y ' i = - d Z/ dx i . . . . . . . . (1) where x 1 ... x n , y 1 ... y n are two sets of n variables each, and accents denote total differentiation with respect to the independent variable t ; Z being any function of x 1 &c., y 1 which may also contain t explicitly. The part now laid before the Society is limited to the consideration of the above form. After deducing from known properties of functional determinants a general theorem to be used afterwards, the author establishes the following propositions.